Faithfully flat ring maps are not descendable

It is an important foundational result in algebraic geometry that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -category of quasi-coherent modules fibered over the category of commutative rings satisfies descent with respect to the fqpc topology. More recently, Akhil Mathew and Jacob Lurie introduced an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -categorical version of this result for module spectra using the notion of <italic> descendable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {E}_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -ring maps </italic> . In this paper, we will construct a faithfully flat algebra over the infinite polynomial ring on an algebraically closed field that is not descendable.